ON THE MOTION OF PENDULUMS. 113 arc of vibration deviated sensibly from that of a geometric progression. In Baily's experiments, only the initial and final arcs are registered, and not even those in the case of the "additional experiments." Hence these experiments do not enable us to make out whether it would be sufficiently exact to suppose the decrease to take place in geometric progression. Moreover, the final arc was generally so small, that a small error committed in the measurement of it would cause a very sensible error in the rate of decrease concluded from the experiment. For these reasons it would be unreasonable to expect a near accordance between the formulae and the results of the experiments of Bessel and Baily. Still, the formulae might be expected to give a result in defect, and yet not so much in defect as not to form a large portion of the result given by observation. On this account it will not be altogether useless to compare theory and observation with reference to the decrement of the arc of vibration. 73. Let us first consider the case of a sphere suspended by a fine wire. Let the notation be the same as was used in investigating the expression for the effect of the air on the time of vibration, except that the factors k', Ictr come in place of k, k . Considering only that part of the resistance which affects the arc of vibration, we have for the portions due respectively to the sphere and to the element of the wire whose length is ds, and distance from the axis of suspension $, ^de 7.,jf/ ^ _ de and if we take the moment of the resistance, and divide by twice clB the moment of inertia, the coefficient of -j- in the result, taken negatively, and multiplied by t, will be the index of e in the expression for the arc. Hence if a0 be the initial arc of vibration, and at the arc at the end of the time t , , a + sl* Trt log. «0 - loge at = ^--j-i——^- . - . . .(108), M (I -f of being as before taken for the moment of inertia of the sphere, which will be abundantly accurate enough. If then we put I for the Napierian logarithm of the ratio of the arc at the It' i S. III.